Math Art
Costa-Hoffman-Meeks Surface

Costa-Hoffman-Meeks Surface

Costa-Hoffman-Meeks Surface is a minimal surface, given by a Weierstrass representation, genus varies, 3 ends, embedded.

Open Costa-Hoffman-Meeks Surface in the interactive viewer →

aperiodic complete-finite-total-curvature def-weierstrass embedded implemented minimal tradition-classical

Formula

Gauss map g
c⁢exp⁡(−(k⁢log⁡(z)+log⁡(1+z)+log⁡(−1+z))1+k)c \exp\left(\frac{-\left(k \log\left(z\right) + \log\left(1 + z\right) + \log\left(-1 + z\right)\right)}{1 + k}\right)
Height differential dh
c−1+z2\frac{c}{-1 + z^{2}}

Properties

Family
minimal
Given by
a Weierstrass representation
Curvature
zero mean curvature
Periodicity
not periodic
Genus
varies
Ends
3
Embedding
embedded
Orientable
yes
Fidelity
exact
Exactness
numerical-integral

Definition

Extracted from the shipped row's source by AST (complete multi-line lambdas, phi triple (dh = phi3, g = phi3/(phi1 - i*phi2), a Weierstrass-representation identity)) and reproduced numerically against the shipped callables over the complex plane, exactly -- no scalar slack.

Sources

Weierstrass data reproduced numerically against the shipped implementation (math_art/minsurf/zoo.py, row COSTA_HM) by sampling g and dh over rings in the complex plane at the row's default parameters: g: matches over 144 complex samples (worst 1.57e-15); dh: matches over 144 complex samples (worst 0).